Aircraft: EMB ERJ 170/175 ER/LR
Phase of flight: Cruise
Anomaly: Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Fuel Issue; Inflight Event / Encounter Weather / Turbulence

[We were] on day three of a five day pairing. We had completed three of our four scheduled legs in challenging weather conditions. Our next leg was to ZZZ to overnight. The weather there was poor and Operations told us that we would likely cancel. I called Dispatch; my Dispatcher was also leaning towards cancellation. They reported weather at ZZZ was 200 feet overcast and 2 sm on a wet runway. He told me to standby and they would make a decision. He called me back to say it was a go and the alternate was ZZZ1. I immediate noticed the poor conditions at ZZZ1 as well. I was sure we needed a 2nd alternate. ZZZ1 reported at the time of our arrival we would be at 500 feet and 3 sm. It was better than that currently but deteriorating. I called back and requested the 2nd alternate. The Dispatcher I could tell was new and he didn't agree with the Supervisor's decision to dispatch to ZZZ and I could tell they were also not excited about my request for a 2nd alternate. They told me we would have to kick off more passengers to accommodate the alternate. I told them it's unfortunate but a must. We received more fuel and they eventually got enough volunteers to go on a later flight. We reviewed the paperwork and decided that our 2nd alternate was a must and ZZZ2 [departure station] was a good one. We took off and headed to ZZZ. We briefed the RNAV and the weather had gone down slightly to 200 feet and 1 sm. We set up the approach [and] while prepping for the approach; Dispatch had sent some confusing messages to us. The first said if you can't make it in to ZZZ1 just return to ZZZ2. I was thinking we didn't have enough fuel for a ZZZ attempt; go missed; shoot ZZZ1 approach; go missed and then go to ZZZ2. During this time; another Dispatcher came on; we were starting the approach into ZZZ and had intercepted course and were about to cross the final approach fix when Tower made a call to us saying that the approach light system shorted out. He did not think they would be available but he said the weather was breaking and he thought we could get in. We went missed [and] climbed to the publish hold. ZZZ1 was dropping; it was around 500 and 3sm; I sent ACARS to Dispatch and she was leaning towards ZZZ1; just like I. However; she was directed to tell us to return to ZZZ2 for Operational concerns. We looked tight on fuel but had enough per their request. We turned 2 turns in the hold; climbed to 10;000 [feet] to save fuel and I immediately asked for ZZZ2. ZZZ1 Center told us that we could expect EFC (Expect Further Clearance) 25 minutes later. We told him it was unacceptable and we were declaring min fuel. He gave us routing; and we told him we could not accept published speeds for fuel economy. Long range cruise was pretty low; but we were in constant rough air so we sped up to give us a margin. I immediately started sending numbers to Dispatch about every fix; (miles out) and FOB (Fuel On Board) to keep her in the loop; and we knew we would be close. ATC was very busy and many aircraft were diverting to ZZZ2; including aircraft inbound to ZZZ1 we were told.By the time we were on final; we had amber gauges; the weather worsened at ZZZ2 and we saw the runway approach lights right at minimums. They had heavy rain at the time. I feel like we should have never been dispatched to an airport that we knew we could not get into. It was suggested that we make ZZZ1 the destination and find another suitable Alternate. My request for that was denied. E175 Captain reported declaring minimum fuel to expedite ATC handling during a divert due to weather at destination.